# Ecole d'Ete de Probabilites de Saint-Flour XV-XVII by P. Diaconis, Ecole D'Ete De Probabilities De, P. L.

By P. Diaconis, Ecole D'Ete De Probabilities De, P. L. Hennequin

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A be any element of * . Let f°° r°° Fn = E[Fn)\ + / Gna dWa, Jo Jo neN, be a sequence in £/& converging to F. Then 7j /•OO + f /J™ JM i a GadWa + y ° ° H a G a d W a \ \\2 |\TF - (riE[F] < 2[\\T(F n)\\ - 2 Fn)\\2 <2[\\T(F-F /•oo \\T1(E[F\ - n})+ E[Fn}) ++ /I A Gna) dWaa+ + \\Tl(E[F\-E[F Ma3(G ( Ga 3 - GJ) Jo 2 , n + j°°° ° . f f . -G )

We will use the sign sum to denote a union of disjoint elements of V. F(yl)| < M^). In the following we always denote by T the kernel of an operator T (if it admits one); in the same way, we also denote by F(A) the coefficients of the chaotic expansion of a vector F of \$ . 4) KF(A)= V / K(U, V, N)F(N + V + W) dN. Finally, for P in V we define VP = max P (with V0 = 0). Ill Integral representation As announced in the introduction we consider a bounded linear operator T from \$ into itself which verifies EtT = T Et, for all t in R+.

2) on £ (6 , for a bounded adapted process H (the terms K dA and L dA' in the decomposition vanish since they can be estimated in terms of the measure). By definition M0u belongs to \$ 0 ] ~ C l , for all u in \$ . So we get M0 = T l 7. It is easy to verify that M< converges strongly to T when t tends to +oo. In other words, TOO T = T1I+ / Jo HadA(s), on £ /6 . That is, for every F G f « , F = E[F] + /0°° G a dW s , we have T F = T1F + / ( M , - T l 7) Gs dW, + / Jo Jo = T l iB[F] + / Jo M3Ga dW3 + / Jo 77SG3 dWs 77SG3 dWs.